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Symbolic powers of prime ideals with applications to hypersurface rings

Published online by Cambridge University Press:  22 January 2016

Sam Huckaba
Sam Huckaba*
Affiliation:
Department of Mathematics Florida State University, Tallahassee, FL 32306-3027, U.S.A.
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Let R be a commutative Noetherian ring and suppose q is a prime ideal of R. A fundamental problem is to decide when powers qn of q are primary (that is qn is its own primary decomposition). If q is generated by a regular sequence then powers of q are always primary, because G(q, R) (the associated graded ring of R with respect to q) is an integral domain (see [12 page 98] and also [5 (2.1)]). Let qn) denote the nth symbolic power of q-defined by q(n) = {rεR|there exists sεR\q such that sr ε qn}. Then qn is primary if and only if qn = q(n) If q is generated by a regular sequence then we call it a complete intersection prime ideal, so if q is a complete intersection prime ideal then qn ≠ q(n) for all n ≥ 1. If q is not a complete intersection then powers need not be primary. If R is a three-dimensional regular local ring and q is a non-complete intersection height two prime ideal for example, then Huneke showed [11 Corollary (2.5)] that qn = q(n) for all n ≥ 2. Thus, for such a prime q it is impossible for qn to occur in the primary decomposition of any ideal. This phenomena increases the difficulty in finding a primary decomposition for an ideal having q as an associated prime.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 161 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

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